Renormalization of the Einstein-Hilbert action

TL;DR

This paper investigates the renormalization of the Einstein-Hilbert action with boundary corners, analyzing conformal anomalies and finite terms in various dimensions, and compares counterterm and Kounterterm methods, including black hole mass calculations.

Contribution

It provides a detailed analysis of boundary corner effects on the renormalization of the Einstein-Hilbert action and compares different renormalization methods in diverse geometric settings.

Findings

Conformal anomaly in even dimensions is independent of boundary corner type.

Finite terms in odd dimensions are independent of corner type when the boundary dimension is odd.

Counterterm and background subtraction methods agree only with the infinite series counterterm expansion.

Abstract

We examine how the Einstein-Hilbert action is renormalized by adding the usual counterterms and additional corner counterterms when the boundary surface has corners. A bulk geometry asymptotic to $H^{d+1}$ can have boundaries $S^k \times H^{d-k}$ and corners for $0\leq k<d$. We show that the conformal anomaly when $d$ is even is independent of $k$. When $d$ is odd the renormalized action is a finite term that we show is independent of $k$ when $k$ is also odd. When $k$ is even we were unable to extract the finite term using the counterterm method and we address this problem using instead the Kounterterm method. We also compute the mass of a two-charged black hole in AdS$_7$ and show that background subtraction agrees with counterterm renormalization only if we use the infinite series expansion for the counterterm.